In this paper, we study the combinatorics of a subcomplex of the Bloch-Kriz cycle complex  used to construct the category of mixed Tate motives. The algebraic cycles we consider properly contain the subalgebra of cycles that correspond to multiple logarithms (as defined in ). We associate an algebra of graphs to our subalgebra of algebraic cycles. We give a purely graphical criterion for admissibilty. We show that sums of bivalent graphs correspond to coboundary elements of the algebraic cycle complex. Finally, we compute the Hodge realization for an infinite family of algebraic cycles represented by sums of graphs that are not describable in the combinatorial language of .